- Email: [email protected]
Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model
ARTICLE IN PRESS
Journal of Biomechanics 40 (2007) 412–426 www.elsevier.com/locate/jbiomech www.JBiomech.com
Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model I. Chatziprodromoua, A. Tricolia, D. Poulikakosa, Y. Ventikosb, a
Laboratory of Thermodynamics in Emerging Technologies, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Zentrum, CH-8092 Zurich, Switzerland b Fluidics and Biocomplexity Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Accepted 14 December 2005
Abstract We have developed a computational simulation model for investigating an often postulated hypothesis connected with aneurysm growth. This hypothesis involves a combination of two parallel and interconnected mechanisms: according to the first mechanism, an endothelium-originating and wall shear stress-driven apoptotic behavior of smooth muscle cells, leading to loss of vascular tone is believed to be important to the aneurysm behavior. Vascular tone refers to the degree of constriction experienced by a blood vessel relative to its maximally dilated state. All resistance and capacitance vessels under basal conditions exhibit some degree of smooth muscle contraction that determines the diameter, and hence tone, of the vessel. The second mechanism is connected to the arterial wall remodeling. Remodeling of the arterial wall under constant tension is a biomechanical process of rupture, degradation and reconstruction of the medial elastin and collagen fibers. In order to investigate these two mechanisms within a computationally tractable framework, we devise mechanical analogues that involve three-dimensional haemodynamics, yielding estimates of the wall shear stress and pressure fields and a quasi-steady approach for the apoptosis and remodeling of the wall. These analogues are guided by experimental information for the connection of stimuli to responses at a cellular level, properly averaged over volumes or surfaces. The model predicts aneurysm growth and can attribute specific roles to the two mechanisms involved: the smooth muscle cell-related loss of tone is important to the initiation of aneurysm growth, but cannot account alone for the formation of fully grown sacks; the fiber-related remodeling is pivotal for the latter. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cerebral aneurysms; Growth model; Computational simulation; Biological fluid mechanics
1. Introduction The mechanics of cerebral saccular aneurysm pathogenesis are not yet well understood. Although different biological mechanisms are identified as possible reasons for the genesis, the growth and the aneurysm rupture, a fully descriptive theory that explains the relevant biomechanical mechanisms cannot be found in the literature. The aim of this study is to investigate computationally a hypothesis that is relating mechanical Corresponding author. Tel.: +44 1865 277944.
E-mail addresses: [email protected] (D. Poulikakos), [email protected] (Y. Ventikos). 0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2005.12.009
and biological responses of the arterial wall, with the hope of shedding some light to a possible biomechanically explained process. The vascular endothelium (VE) comprises the inner surface of blood vessels and its main role is to be a selective barrier between the blood and the rest of the biological tissues, as well as a wall shear stress sensor. The reaction of the vascular endothelium to blood flow was observed more than 150 years ago; Virchow identified that the morphology of the endothelial cells is varying along the arterial tree due to the variability of the corresponding blood flow patterns (Resnick et al., 2003). Since then, the underlying mechanisms that are responsible for the endothelial cell (EC) function
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
became an area of interest for many investigators. In this direction, a lot of different aspects, such as Ca++ and K+ ion channels activation (Adams et al., 1989; Ando et al., 1993), gene expression differentiation due to shear forces (Volin et al., 1998; Resnick et al., 1997; Shyy et al., 1994), mechanotransduction and mechanosignaling events (Davies et al., 1997; Helmke and Davies, 2002), reorganization and EC alignment (Malek and Izumo, 1996; Satcher et al., 1992), have been studied in depth for the past three decades. One of the most important tasks of the VE is to regulate the arterial wall properties through a blood shear stress activation mechanism. Luis J. Ignarro in his Nobel Prize lecture summarized the steps taken that led to the identification of NO as a principal endothelium derived relaxation factor (EDRF) (Ignarro, 1999). This endogenous molecule alters the properties of the arterial wall by relaxing the smooth muscle cells (SMC) embedded in the arterial media layer. A series of studies have shown that NO production is positively correlated with the changes of the shear stress (SS) values on the arterial wall through an endothelium-dependent biological mechanism (Buga et al., 1991; Tronc et al., 1996; Snow et al., 2001; Wilkinson et al., 2002; Butler et al., 2000; Kinlay et al., 2001). In our aneurysmal genesis hypothesis, this correlation plays a key role for the initialization of the lesion. The quantification of clinically important haemodynamic variables, such as blood velocity, pressure and shear stress, can provide information for the healthy and diseased cardiovascular function. There are numerous experimental and computational methods that are targeting the estimation of such quantities. In practice, parameters like shear stress and blood flow rate are derived indirectly from the primary parameters (pressure and velocity) that are measured/computed directly. Experimental methods can be divided in in-vitro methods (where phantoms are used) and in-vivo methods (where the experiments are conducted directly on human or animal subjects). In-vitro methods are considerably easier to employ, because they do not have the ethical and technological restrictions that in-vivo methods inevitably entail. Catheters equipped with tip pressure transducers are the main instrument for pressure measurements for both in-vitro (Millar and Baker, 1973) and in-vitro (Debruyne et al., 1994) experiments. As far as flow evaluation is concerned, a large number of qualitative and quantitative experimental techniques exist, like hydrogen bubble visualization (Zarins et al., 1983), transit time ultrasound flow probes (Tabrizchi and Pugsley, 2000; Molloi et al., 1996; Molloi et al., 2004), and hot wire velocimetry (Nerem and Seed, 1972). Advances in technology allowed the development of methods that are not invasive (or are minimally invasive). Such methods include computer tomography (CT), ultrasound and magnetic resonance
413
imaging (MRI). Ultrasound and MRI represent the current and prevailing methods in in vivo blood velocimetry but are still not always able to produce dependable and accurate measurements, due to motion artifacts, temporal and spatial resolution limitations, etc. Despite current limitations, these techniques show great promise for providing subject-specific details on important haemodynamic quantities. MRI and Ultrasound techniques find a complimentary counterpart in a modeling methodology developed in recent years, namely Computational Haemodynamics. The main advantages of computational techniques, is that they are non-invasive, they are fully repeatable and they are often capable of providing us with insightful information that is not accessible by any other means. The computational efforts went through a number of steps and advanced in parallel with the computer technology evolution. The result of this parallel evolution produced methods that are ranging from 0D to 3D models (Sherwin et al., 2000), idealized or realistic geometries, with deformable and/or collapsible (fluid-structure interaction) (Otis et al., 1993; Bertram and Pedley, 1982; Perktold and Rappitsch, 1995) or rigid walls and Newtonian or non-Newtonian (Ballyk et al., 1994) approximation for the blood. More specifically, as far as 0D and 1D models are concerned, lamped capacity models, making use of an analogy between fluid flow and electric circuits or other computer-time effective techniques (Westerhof and Noordergraaf, 1970; Wang and Parker, 2004), have yielded useful information. More detailed local information started becoming available with 2D (Steinman et al., 1993) and 3D models (Perktold et al., 1998; Rappitsch et al., 1997; Steiger and Perktold, 1997; Boutsianis et al., 2004) that resulted in a more descriptive perspective of the flow field. Furthermore, the MRI technology makes possible more sophisticated calculations within realistic and patient-specific arterial geometries (Butty et al., 2002; Steinman, 2002; Steinman et al., 2002; Chatziprodromou et al., 2003; Taylor and Draney, 2004). As mentioned before, computational techniques can be closely connected with Ultrasound or MRI blood velocimetry, since these methods provide boundary conditions for patient-specific haemodynamics. Finally, a topic that has drawn the attention of the bioengineering community, of interest to the present study, is the exploration of the structural properties and behavior of the arterial wall. In this direction, both experimental and computational methods have been published. Arterial residual stresses (Fung, 1991; Fung and Liu, 1992), elastic properties and behavior of the various arterial layers (Jogestrand et al., 2003) and mathematical modeling of the arterial function (SchulzeBauer and Holzapfel, 2003; Holzapfel et al., 2002; Gasser and Holzapfel, 2003) are some of the aspects that
ARTICLE IN PRESS 414
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
the scientists are focusing their efforts. A comprehensive review of continuum biomechanics for soft biological tissues can be found in Humphrey (2003). The two important biological mechanisms upon which our hypothesis is based are the apoptosis of the medial smooth muscle cells (Kondo et al., 1998; Schmid et al., 2003; Thompson et al., 1997; Sakaki et al., 1997) and the breakage and elimination of the collagen and elastin fibers within the aneurysmal wall (Mimata et al., 1997; Gaetani et al., 1998; Carmo et al., 2002; Finlay et al., 1995). The present work explores a hypothesis related to the inception and growth of saccular cerebral aneurysms. Our hypothesis is summarized as follows: Hypothesis. A VE malfunction or/and an abnormal shear stress field due to the presence of atherosclerotic plaque or an arterial bifurcation, can cause an overexpression of the Endothelium dependent NO production which leads to smooth muscle cell relaxation and consequently to lower, non-physiological local arterial aggregate Young’s Modulus of Elasticity. This process results in a disturbance of the equilibrium between the blood pressure forces and the internal wall stress forces in favor of the first and subsequently dilates locally the arterial wall causing an initial ballooning. The resulting blood shear stress field—after the above-described geometrical development—in conjunction with a possible thinning of the medial layer because of the smooth muscle cell apoptotic mechanism, are the driving forces for further growth of the aneurysmal geometry. This geometrical growth stretches the collagen and elastin fibers of the medial and adventitial layers and gives rise to internal stresses that contribute to the arterial stiffness. Eventually, the biomechanical system equilibrates at a state where the internal wall stresses and the transmural pressure are equal whilst the local haemodynamics cannot alter the arterial properties any more. At this point, the elastin and the collagen fibers that are responsible for the internal stresses are constantly under a non-physiological, large mechanical load. This tensile state of the above mentioned fibers gradually causes their degradation and breakage and results in a remodeling of the wall and consequently in a new vascular tone. In this new state, the artery is again vulnerable to the pressure field, since we have a new equilibrium and further aneurysmal growth, or rupture, is possible. In order to investigate this hypothesis, we have developed a model that includes the luminal haemodynamics and the tissue response within a computationally tractable framework. Inevitably, a number of assumptions are made, especially with respect to the extremely complicated and poorly understood biochemistry and biomechanics of the processes discussed. The model will be presented in the following section. Subsequently the
results from its application to an idealized aneurysm growth study will be discussed.
2. Modeling and computational methods The core of the model involves a quasi-steady approach, supported by the disparate scales of the local time-accurate haemodynamics, O (s), and the evolution of an aneurysm, O (months) or more. The approach entails the estimation of a prevalent haemodynamic state and subsequently the estimation of the evolution of the disease under the assumption that this state remains constant, until geometric considerations dictate a reevaluation of the haemodynamics. In this manner, we have the flexibility to account for the haemodynamics in a variety of ways. In the present study, and for expedience, we have conducted all the step-wise quasisteady computations of the haemodynamic behavior in a steady (average flow) manner. This approach serves in demonstrating the key features of the model, which lie in the decoupling of the scales and in the coupling of the mechanics with the cellular responses, but it is not restrictive. An idealized geometry of a common carotid artery was used for the application of the computational model that mimics the process described in our hypothesis. The geometry was constructed with the use of a CAD modeler/grid generator (Gambit, FLUENT INC.), Fig. 1(a). This initial geometry was used for a stress pre-evaluation step: Arteries at normal equilibrium conditions have residual stresses in the arterial walls, because of the blood flow and the pressure of the blood they contain. In order to model these stresses, we performed an initial calculation, in a fully coupled fluidstructure interaction manner that was equivalent with filling an unstressed (i.e. empty) artery with blood at physiological flow conditions. The resulting geometry of this initial computation with the calculated internal stresses was used as the base for all further calculations (Fig. 1(b)). This first step caused an increase of the initial diameter of the common carotid artery from 6.5 mm and a corresponding Reynolds number of 200 to a diameter of 8 mm and a Reynolds equal to 165. The thickness of the arterial wall was 0.75 mm and the length of the tubular model was 65 mm. The final dimensions and flow conditions achieved are intended to be typical for carotids. A Young’s modulus of Elasticity of 120 000 Pa and a Poisson ratio of 0.4 were used for this initial step. For the blood, a Newtonian behavior assumption was made (Perktold et al., 1991; Cho and Kensey, 1991). Constant density of 1069 kg/m3 and dynamic viscosity coefficient of 0.0035 (Kg/m s) were the blood physical parameters. The boundary conditions employed were uniform velocity equal to 0.1 m/s for the inlet and fixed physiological pressure 6665 Pa (50 mm Hg) for the
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
415
Fig. 1. Flow and Stress Boundary Conditions and resulting geometry after the internal stress pre-evaluation step.
Fig. 3. Volume region nomenclature.
Fig. 2. Wall Strain rate on a line along the X-axis (vessel axial line) that passes from the top of the aneurysm for the time t6a (the most distorted geometry computed). Two grids compared: 395094 elements and 833975 elements.
outlet. These boundary conditions correspond to a characteristic flow setting for typical carotid arteries. The boundary conditions that were used for the solid problem were fixed zero displacement for the face adjacent to the inflow and outflow sections and a free boundary condition for the outer surface of the arterial wall. The coupling of the fluid mechanics and the solid deformations was achieved through an implicit pressure boundary condition at the luminal arterial wall. After this initial step was completed, the general quasi-steady algorithm was employed to study the growth of an induced aneurysm in this pre-stressed arterial segment. This algorithm can be summarized as follows: 1. An adequate (after a grid independence study, Fig. 2) unstructured tetrahedral grid was constructed. The grid consisted of 395094 elements for the flow
solution computations and of 915179 elements for the fluid-structure interaction runs (444223 elements for the fluid volume—470956 elements for the solid volume). At each time step, the mesh was refined and the element size kept constant, in spite of the deformations in the geometry. The grid was generated using a mesh generator (Gambit, Fluent) for the initial regional constituents. 2. A finite volume solver was employed for the numerical solution of the flow within the parent vessel and the aneurysm (CFD–ACE+, ESI CFD). A second order accurate discretization scheme was used, along with a SIMPLEC-type pressure-velocity coupling scheme. Algebraic multigrid acceleration was employed in all simulations in order to enhance the speed of convergence of the solution. The blood flow equations that were solved by the finite volume solver are the incompressible Navier–Stokes equations along with continuity, i.e. the conservation laws for momentum and mass for incompressible fluids. The conservation of mass can be expressed as @ui ¼ 0. @xi
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
416
Conservation of momentum reads: r
@ui @ui @p @tij þ ruj ¼ þ þ SMx . @xi @xj @t @xj
The viscous stress components tij read: 1 @ui @uj tij ¼ 2msij where sij ¼ þ 2 @xj @xi for Newtonian flows with constant viscosity m. 3. Through a fluid-structure interaction process, the flow-induced forces that were computed act as an Table 1 Geometrical data for the two volumes where the geometrical evolution took place
Ellipsoidal Ring shaped
R1 (m, big axis of the ellipse along the length of the tube)
R2 (m, small axis of the ellipse perpendicular to the length of the tube)
0.0035 0.0045
0.002 0.003
implicit force on the arterial wall. The resulting wall deformation was calculated using a finite element stress solver (CFD–ACE+, ESI CFD). This fluidstructure interaction computation is based on the strong implicit coupling between the solid-stress strain computation and the fluid computation. Pressure and shear stress information is sent to the stress computation module, where deformations and stresses are calculated. Subsequently, these deformations are fed back to the fluid module, where a new grid is generated and the solution is recalculated on this new deformed geometry. Iterations are performed until convergence is obtained (i.e. the residuals for the three velocity components and for the pressure dropped nine orders of magnitude and stabilized). The structural part of this algorithm is based on the Finite Element method. The principle of virtual work (Zienkiewicz, 1971) is used for the implementation of the stress-deformation computation. For each element, displacements are defined at the nodes and obtained within the element by interpolation from the nodal values using appropriate
Fig. 4. Geometrical evolution contours due to the Young’s Modulus of Elasticity dynamic alteration on the longitudinal symmetry plane.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
shape functions. Using the strain–displacement relationship, the strains e are derived from the displacements u, hence the nodal displacements a. This may be expressed as de ¼ Bda. If the displacements are large, the strains depend nonlinearly on the displacements, thus B is a function of a. This is expressed as B ¼ B0 þ BL ðaÞ, where B0 is the standard small-strain strain–displacement matrix, and BL is a linear function of the nodal displacement. The governing equations are derived by forming a balance between the external and internal generalized forces using the principal of virtual work mentioned above. Equating the external work done with the total internal work, and recognizing this equality must be valid for any value of virtual displacement, we reach an equilibrium equation which for the general nonlinear case is solved iteratively via a Newton–Raphson technique. We must note that, although the general elasticity relationship, s ¼ Dðe2e0 Þ2s0 , is used, this approach is sufficiently general to allow for any nonlinear stress–strain relationship, even the
417
variable exponential ones that prevail in artery wall dynamics, since the solution will again reduce to the solution of a set of nonlinear equations, (Bathe and Wilson, 1976; Newmark, 1959; Roark and Young, 1975). We approximated the arterial wall as a fully elastic medium, with the above-mentioned characteristics. 4. The above step causes a geometrical evolution and a subsequent grid deformation both for the fluid and the solid areas. Smooth and controlable deformation of the luminal zone was derived by solving a pseudostress-strain problem for the fluid volume, geared exclusively towards evaluating the new position of the internal nodes under the new boundary node configuration. When the deformed mesh proved to be of sufficient quality for the re-calculation of the flow within the new geometry, that computation took place in a straightforward manner. In situations were the deformation was too large, remeshing of the luminal volume was conducted, to satisfy the abovementioned constraint.
Fig. 5. Geometrical evolution contours due to the Young’s Modulus of Elasticity dynamical alteration up to time t5 and the arterial adaptation that is taking place at time t6a.
ARTICLE IN PRESS 418
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
The geometrical evolution was carried out in eight quasi-steady steps. For inducing the inception of an aneurysm-prone area, a simplified two-zonal differentiation was imposed to the wall properties: An ellipsoidal (EL) and a ring-shaped (RS) zone were introduced,
(Fig. 3 and Table 1). At these regions, the strength of the artery (i.e. YME) was decreased linearly (due to the lack of quantitative data) during the computation of the aneurysm evolution. This linear dynamic YME alteration is used to model the above described endothelial cell
Fig. 6. Wall Shear Stress patterns in (Pa) for the eight time instances at the aneurysm region (top view).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
layer function and the corresponding smooth muscle cell degeneration which is derived from a combination of information found in numerous, predominantly in vitro, investigations (Ignarro, 1999; Buga et al., 1991; Tronc
419
et al., 1996; Snow et al., 2001; Wilkinson et al., 2002; Butler et al., 2000; Kinlay et al., 2001). This change in the YME for the EL was in the range from 120 000 to 20 000 Pa, while the corresponding line for the RS
Fig. 7. Pressure patterns in (Pa) for the eight time instances at the aneurysm region (top view).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
420
ranged between the values 120 000 and 40 000 Pa. The reasoning for this differentiation of the two regions stems from the fact that the aneurysmal neck has been shown to develop significantly higher values of WSS (Tronc et al., 1996; Gnasso et al., 2001). The above-mentioned eight consecutive steps of wall weakening allow us to draw interesting conclusions on the plausibility of the mechanical wall-weakening hypothesis, as we shall see in the next section. To fully explore our basic hypothesis, one additional mechanism has to be incorporated though: the effect of arterial wall remodeling under constant load, a mechanism is of major importance as we shall see. There is inconclusive evidence in the literature as to whether this process occurs rapidly (always referring to disease evolutiontime terms) or gradually. For this study, we decided to simulate this behavior under the (strong) assumption that the aneurysmal wall fully remodels itself to an unstressed state within one simulated disease evolution step. The reason for this assumption is that we would like at this stage to point out the importance of the remodeling to the aneurysmal evolution. The abruptness involved in the rupture of individual collagen fibers (Mimata et al., 1997; Gaetani et al., 1998; Carmo et al., 2002; Finlay et al., 1995) (a process that in principle justifies our assumption) is smoothened out in time by the fact that there are numerous fibers undergoing this rupture and re-adjustment of tensile state. The stress readjustment calculation was performed on the geometry that was resulted from the 5th disease evolution step (t5) calculation.
3. Results and discussion Fig. 4 shows the evolution of the aneurysmal shape only due to the dynamic alteration of the Young’s modulus of elasticity. If we consider that the final YME values for time t ¼ 9 of the ring- and the ellipsoidshaped regions are highly abnormal (20 000 and 40 000 Pa, respectively) when at the same time the displacement is not so significant, (compared to cases of giant aneurysms, (Vinuela et al., 1997)) we can conclude that this mechanism alone is not enough to
describe the aneurysmal evolution. In all likelihood, this mechanical process is reflecting a primary malfunction of the endothelial cell layer and the corresponding loss of vascular tone. Moreover, although there exists evidence of apoptosis of the smooth muscle cells during the growth of the aneurysm, we do not know why and how exactly this takes place, (Kondo et al., 1998; Schmid et al., 2003; Thompson et al., 1997; Sakaki et al., 1997). Definitive evidence that the endothelial cell activity is directly connected with the apoptotic processes is missing. In effect, this loss of tone is assumed to account for the initialization of the lesion. This comment is in agreement with the observed high concentration of endothelial cells at high shear stress regions (Hoshina et al., 2003) while histological data indicate that the endothelial cell layer within the aneurysmal lumen is quite normal (Sugiu et al., 1995). This conclusion would indicate that, while we have an unusually high concentration of endothelial cells at the early stages of the decease, at the later stages we have a normal alignment of the cells with respect to the inner layer of the aneurysmal lumen. The YME dynamical evolution up to time t ¼ 5, along with the arterial adaptation mechanism (i.e. canceling of the internal stresses values and establishment of a stress- free state for the resulted geometry from the t ¼ 5 simulation) that took place at time 6a resulted in the sequence of the geometries that are depicted in Fig. 5. It is obvious that the resulting geometry for t ¼ 6a is significantly more deformed than the one achieved by the simple loss of tone/apoptosis mechanism. The exact magnitude of this particular deformation is of course also dependent on the fact that a total loss of stresses was assumed in the fibers of the wall during this step, an assumption that, as we Table 3 Total integral pressure force on the various segments of the inner (in) surface of the arterial wall (N)
Fine
Stretched
Wall in Ring in Elips. in
0.370001 0.175 0.194
0.370001 0.175 0.194
Table 2 Integral pressure force components on the various segments of the inner (in) surface of the arterial wall (N)
X
Y
Fine Wall in Ring in Elips. in
Stretched 4
6.92 10 6.37 105 1.41 104
Z
Fine 4
7.05 10 6.36 105 1.41 104
Stretched 1
3.70 10 1.75 101 1.94 101
Fine 1
3.70 10 1.75 101 1.94 101
Stretched 5
5.69 10 5.49 105 1.77 106
5.68 105 5.44 105 2.05 106
Stretched grid: The grid that results from the grid deformation due to the geometrical evolution. Fine grid: The grid-independent regenerated mesh that was used for the blood flow calculation.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
discussed, is useful for demonstration purposes but possibly not entirely realistic. The influence and the importance of the adaptation procedure will be examined in the future, in correlation with availability of micro-histology data. The Wall Shear Stress patterns at the various stages during the aneurysmal growth are shown in Fig. 6. These results coincide and justify the observation that the shear stress presents with higher values at the neck neighborhood, hence, the YME is changing more than the corresponding value at the center of the lumen at each time step (Banerjee et al., 1996). Essentially, mean WSS values identified in this step are the driving stimuli that dictate the variation of the YME of the corresponding layers of the arterial wall. The bi-zonal approach we
421
developed is also justified by the average observed behavior of the WSS in these figures. Fig. 7 shows that the pressure field on the wall remains practically constant for the entire domain. This result, together with the fact that the pressure field depicted is obtained after the blood lumen grid is adequately refined, confirms the statement that the coupled flow-stress calculation is sufficiently resolved for driving the process (Tables 2 and 3). The streamline (color-coded with the velocity magnitude) comparison between the t ¼ 6 and 6a case is presented in Fig. 8. The importance of the arterial wall adaptation mechanism and its influence on the arterial haemodynamics is further accentuated by this comparison. The latter case leads to a severely deformed geometry faster; the flow field
Fig. 8. Wall Shear Stress and Streamline comparison for the corresponding geometries at the time instances t6 and t6a (resulting geometry after the wall remodelling and the ballooning from the fluid—structure interaction).
ARTICLE IN PRESS 422
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
within the aneurysmal sac is much more disturbed at this stage. This in turn results in a wall shear stress spatial distribution of more irregular patterns and the possibility of further activation of the endothelial cell function.
Since the arterial wall stress tensor has six components and it is difficult to present any illustrative results based on it, the energetic equivalent formulation by von Mises is used to depict the stress history of the arterial
Fig. 9. Von Mises stresses in (N/m2) for the eight time instances at the aneurysm region on the inner surface of the arterial wall.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
wall during its deformation process. The equivalent von Mises stress characterizes the distortional energy (s2/E) within an element and gives a stress value in terms of equivalent positive defined scalar stress levels. In Figs. 9 and 10 contours of the above-mentioned quantity are shown, for the luminal and adventitial surfaces of the arterial wall respectively. The symmetry
423
of the geometry is reflected on the von Mises stress values for both the inner and the outer surfaces. The total absence of initial internal stresses for the wall remodeling case causes an abrupt and rapid deformation (Fig. 11). The difference between the final stresses of the remodeled and the non-remodeled case is clear. This difference is a result of our decision to eliminate the
Fig. 10. Von Mises stresses in (N/m2) for the eight time instances at the aneurysm region on the outer surface of the arterial wall.
ARTICLE IN PRESS 424
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
internal stresses in order to underline the importance of the remodeling procedure. This result gives the impression that the bulk part of the deformation is caused by this remodeling procedure and more or less the shaping of the aneurysm is the result of an one step procedure. From Fig. 11, we can also see that the resulting spatial stresses from this deformation are similar to that during the inflation of a balloon. Moreover, because of the large deformation near the apex of this ‘‘balloon’’, we are finding the largest stress values for the outer wall at the territory with the higher YME value.
4. Conclusions We have developed a computational model for the growth of saccular cerebral aneurysms. The model relies
on certain simplifying assumptions: the bizonal approach, and the relation of smooth muscle cell tone loss and apoptosis with the WSS patterns are two of the most important ones. Moreover, for the purposes of the present study, we have limited the driving haemodynamics to steady state simulations. On the other hand, correlating accurately detailed quantitative data of vasoactive agent overproduction with local wall shear stress patterns involves detailed in vitro or better yet in vivo experiments which, to the best of our knowledge, do not exist. With the above approximations, the model developed shows that the combination of two individual mechanisms, often postulated in the literature as responsible for aneurysm growth, are indeed of importance to the process: Loss of vascular tone due to smooth muscle cell apoptosis and reconstitution of fibers after prolonged periods of excessive strain have
Fig. 11. Von Mises stresses (N/m2) comparison on the inner and the outer surfaces of the arterial wall, between the corresponding geometries at the time points t6 and t6a (resulting geometry after the wall remodelling and the ballooning from the fluid—structure interaction).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
been shown to contribute to the condition. Moreover, we have provided evidence that although the first mechanism is important to the inception of the disease, for substantial aneurysm growth to occur, the second mechanism is pivotal.
Acknowledgements The authors are grateful to the ETH Forschungscommission for supporting this work. The ESI Group and CFD–RC (Dr. S. Sundaram) are kindly acknowledged for their advice and support and for allowing the usage of the CFD–ACE platform. One of the authors (YV) would like to acknowledge discussions with Dr. Robin Poston of King’s College, London, that affirmed further our causal hypothesis for the disease, based on pathological and histological data, during the revision of this paper. References Adams, D.J., Barakeh, J., Laskey, R., Vanbreemen, C., 1989. Ion channels and regulation of intracellular calcium in vascular endothelial-cells. Faseb Journal 3 (12), 2389–2400. Ando, J., Ohtsuka, A., Korenaga, R., Kawamura, T., Kamiya, A., 1993. Wall shear-stress rather than shear rate regulates cytoplasmic Ca++ responses to flow in vascular endothelial-cells. Biochemical and Biophysical Research Communications 190 (3), 716–723. Ballyk, P.D., Steinman, D.A., Ethier, C.R., 1994. Simulation of nonnewtonian blood-flow in an end-to-side anastomosis. Biorheology 31 (5), 565–586. Banerjee, R.K., Gonzalez, C.F., Cho, Y.I., Picard, L., 1996. Hemodynamic changes in recurrent intracranial terminal aneurysm after endovascular treatment. Academic Radiology 3 (3), 202–211. Bathe, K., Wilson, E.L., 1976. Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, New Jersey. Bertram, C.D., Pedley, T.J., 1982. A mathematical-model of unsteady collapsible tube behavior. Journal of Biomechanics 15 (1), 39–50. Boutsianis, E., Dave, H., Frauenfelder, T., Poulikakos, D., Wildermuth, S., Turina, M., Ventikos, Y., Zund, G., 2004. Computational simulation of intracoronary flow based on real coronary geometry. European Journal of Cardio-Thoracic Surgery 26 (2), 248–256. Buga, G.M., Gold, M.E., Fukuto, J.M., Ignarro, L.J., 1991. Shearstress induced release of nitric-oxide from endothelial-cells grown on beads. Hypertension 17 (2), 187–193. Butler, P.J., Weinbaum, S., Chien, S., Lemons, D.E., 2000. Endothelium-dependent, shear-induced vasodilation is rate- sensitive. Microcirculation 7 (1), 53–65. Butty, V.D., Gudjonsson, K., Buchel, P., Makhijani, V.B., Ventikos, Y., Poulikakos, D., 2002. Residence times and basins of attraction for a realistic right internal carotid artery with two aneurysms. Biorheology 39 (3–4), 387–393. Carmo, M., Colombo, L., Bruno, A., Corsi, F.R.M., Roncoroni, L., Cuttin, M.S., Radice, F., Mussini, E., Settembrini, P.G., 2002. Alteration of elastin, collagen and their cross-links in abdominal aortic aneurysms. European Journal of Vascular and Endovascular Surgery 23 (6), 543–549. Chatziprodromou, I., Butty, V.D., Makhijani, V.B., Poulikakos, D., Ventikos, Y., 2003. Pulsatile blood flow in anatomically accurate vessels with multiple aneurysms: a medical intervention planning
425
application of computational haemodynamics. Flow Turbulence and Combustion 71 (1–4), 333–346. Cho, Y.I., Kensey, K.R., 1991. Effects of the non-newtonian viscosity of blood on flows in a diseased arterial vessel.1. Steady flows. Biorheology 28 (3–4), 241–262. Davies, P.F., Barbee, K.A., Volin, M.V., Robotewskyj, A., Chen, J., Joseph, L., Griem, M.L., Wernick, M.N., Jacobs, E., Polacek, D.C., DePaola, N., Barakat, A.I., 1997. Spatial relationships in early signaling events of flow-mediated endothelial mechanotransduction. Annual Review of Physiology 59, 527–549. Debruyne, B., Stockbroeckx, J., Demoor, D., Heyndrickx, G.R., Kern, M.J., 1994. Role of side holes in guide catheters—observations on coronary pressure and flow. Catheterization and Cardiovascular Diagnosis 33 (2), 145–152. Finlay, H.M., McCullough, L., Canham, P.B., 1995. 3-Dimensional collagen organization of human brain arteries at different transmural pressures. Journal of Vascular Research 32 (5), 301–312. Fung, Y.C., 1991. What are the residual-stresses doing in our bloodvessels. Annals of Biomedical Engineering 19 (3), 237–249. Fung, Y.C., Liu, S.Q., 1992. Strain distribution in small blood-vessels with zero-stress state taken into consideration. American Journal of Physiology 262 (2), H544–H552. Gaetani, P., Tartara, F., Grazioli, V., Tancioni, F., Infuso, L., Baena, R.R.Y., 1998. Collagen cross-linkage, elastolytic and collagenolytic activities in cerebral aneurysms: a preliminary investigation. Life Sciences 63 (4), 285–292. Gasser, T.C., Holzapfel, G.A., 2003. Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Computer Methods in Applied Mechanics and Engineering 192 (47–48), 5059–5098. Gnasso, A., Carallo, C., Irace, C., De Franceschi, M.S., Mattioli, P.L., Motti, C., Cortese, C., 2001. Association between wall shear stress and flow-mediated vasodilation in healthy men. Atherosclerosis 156 (1), 171–176. Helmke, B.P., Davies, P.F., 2002. The cytoskeleton under external fluid mechanical forces: Hemodynamic forces acting on the endothelium. Annals of Biomedical Engineering 30 (3), 284–296. Holzapfel, G.A., Gasser, T.C., Stadler, M., 2002. A structural model for the viscoelastic behavior of arterial walls: continuum formulation and finite element analysis. European Journal of Mechanics aSolids 21 (3), 441–463. Hoshina, K., Sho, E., Sho, M., Nakahashi, T.K., Dalman, R.L., 2003. Wall shear stress and strain modulate experimental aneurysm cellularity. Journal of Vascular Surgery 37 (5), 1067–1074. Humphrey, J.D., 2003. Continuum biomechanics of soft biological tissues. Proceedings of the Royal Society of London Series aMathematical Physical and Engineering Sciences 459 (2029), 3–46. Ignarro, L.J., 1999. Nitric oxide: a unique endogenous signaling molecule in vascular biology (Nobel lecture). Angewandte ChemieInternational Edition 38 (13–14), 1882–1892. Jogestrand, T., Eiken, O., Nowak, J., 2003. Relation between the elastic properties and intima-media thickness of the common carotid artery. Clinical Physiology and Functional Imaging 23 (3), 134–137. Kinlay, S., Creager, M.A., Fukumoto, M., Hikita, H., Fang, J.C., Selwyn, A.P., Ganz, P., 2001. Endothelium-derived nitric oxide regulates arterial elasticity in human arteries in vivo. Hypertension 38 (5), 1049–1053. Kondo, S., Hashimoto, N., Kikuchi, H., Hazama, F., Nagata, I., Kataoka, H., 1998. Apoptosis of medial smooth muscle cells in the development of saccular cerebral aneurysms in rats. Stroke 29 (1), 181–188. Malek, A.M., Izumo, S., 1996. Mechanism of endothelial cell shape change and cytoskeletal remodeling in response to fluid shear stress. Journal of Cell Science 109, 713–726.
ARTICLE IN PRESS 426
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
Millar, H.D., Baker, L.E., 1973. Stable ultraminiature catheter-tip pressure transducer. Medical & Biological Engineering 11 (1), 86–89. Mimata, C., Kitaoka, M., Nagahiro, S., Iyama, K., Hori, H., Yoshioka, H., Ushio, Y., 1997. Differential distribution and expressions of collagens in the cerebral aneurysmal wall. Acta Neuropathologica 94 (3), 197–206. Molloi, S., Ersahin, A., Tang, J., Hicks, J., Leung, C.Y., 1996. Quantification of volumetric coronary blood flow with dual- energy digital subtraction angiography. Circulation 93 (10), 1919–1927. Molloi, S., Zhou, Y.F., Kassab, G.S., 2004. Regional volumetric coronary blood flow measurement by digital angiography: in vivo validation. Academic Radiology 11 (7), 757–766. Nerem, R.M., Seed, W.A., 1972. In-vivo study of aortic flow disturbances. Cardiovascular Research 6 (1), 1–14. Newmark, N.M., 1959. A method of computation for structural dynamics. ASCE Journal of the Engineering Mechanics Division 85, 67–94. Otis, D.R., Johnson, M., Pedley, T.J., Kamm, R.D., 1993. Role of pulmonary surfactant in airway-closure—a computational study. Journal of Applied Physiology 75 (3), 1323–1333. Perktold, K., Rappitsch, G., 1995. Computer-simulation of local blood-flow and vessel mechanics in a compliant carotid-artery bifurcation model. Journal of Biomechanics 28 (7), 845–856. Perktold, K., Peter, R.O., Resch, M., Langs, G., 1991. Pulsatile nonnewtonian blood-flow in 3-dimensional carotid bifurcation models—a numerical study of flow phenomena under different bifurcation angles. Journal of Biomedical Engineering 13 (6), 507–515. Perktold, K., Hofer, M., Rappitsch, G., Loew, M., Kuban, B.D., Friedman, M.H., 1998. Validated computation of physiologic flow in a realistic coronary artery branch. Journal of Biomechanics 31 (3), 217–228. Rappitsch, G., Perktold, K., Pernkopf, E., 1997. Numerical modelling of shear-dependent mass transfer in large arteries. International Journal for Numerical Methods in Fluids 25 (7), 847–857. Resnick, N., Yahav, H., Khachigian, L.M., Collins, T., Anderson, K.R., Dewey, F.C., Gimbrone, M.A., 1997. Endothelial gene regulation by laminar shear stress. In: Analytical and Quantitative Cardiology, pp. 155–164. Resnick, N., Yahav, H., Shay-Salit, A., Shushy, M., Schubert, S., Zilberman, L.C.M., Wofovitz, E., 2003. Fluid shear stress and the vascular endothelium: for better and for worse. Progress in Biophysics & Molecular Biology 81 (3), 177–199. Roark, R.J., Young, W.C., 1975. In: M.-H.B. Co. (Ed.), Formulas for Stress and Strain, New York. Sakaki, T., Kohmura, E., Kishiguchi, T., Yuguchi, T., Yamashita, T., Hayakawa, T., 1997. Loss and apoptosis of smooth muscle cells in intracranial aneurysms—studies with in situ DNA end labeling and antibody against single-stranded DNA. Acta Neurochirurgica 139 (5), 469–474. Satcher, R.L., Bussolari, S.R., Gimbrone, M.A., Dewey, C.F., 1992. The distribution of fluid forces on model arterial endothelium using computational fluid-dynamics. Journal of Biomechanical Engineering—Transactions of the ASME 114 (3), 309–316. Schmid, F.X., Bielenberg, K., Schneider, A., Haussler, A., Keyser, A., Birnbaum, D., 2003. Ascending aortic aneurysm associated with bicuspid and tricuspid aortic valve: involvement and clinical relevance of smooth muscle cell apoptosis and expression of cell death- initiating proteins. European Journal of Cardio-Thoracic Surgery 23 (4), 537–543. Schulze-Bauer, C.A.J., Holzapfel, G.A., 2003. Determination of constitutive equations for human arteries from clinical data. Journal of Biomechanics 36 (2), 165–169. Sherwin, S.J., Shah, O., Doorly, D.J., Peiro, J., Papaharilaou, Y., Watkins, N., Caro, C.G., Dumoulin, C.L., 2000. The influence of
out-of-plane geometry on the flow within a distal end-to-side anastomosis. Journal of Biomechanical Engineering-Transactions of the Asme 122 (1), 86–95. Shyy, Y.J., Hsieh, H.J., Usami, S., Chien, S., 1994. Fluid Shear-Stress Induces a Biphasic Response of Human Monocyte Chemotactic Protein-1 Gene-Expression in Vascular Endothelium. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 91(11). pp. 4678–4682. Snow, H.M., Markos, F., O’Regan, D., Pollock, K., 2001. Characteristics of arterial wall shear stress which cause endotheliumdependent vasodilatation in the anaesthetized dog. Journal of Physiology—London 531 (3), 843–848. Steiger, H.J., Perktold, K., 1997. Computer modeling of intracranial saccular and lateral aneurysms for the study of their hemodynamics. Neurosurgery 41 (1), 326. Steinman, D.A., 2002. Image-based computational fluid dynamics modeling in realistic arterial geometries. Annals of Biomedical Engineering 30 (4), 483–497. Steinman, D.A., Vinh, B., Ethier, C.R., Ojha, M., Cobbold, R.S.C., Johnston, K.W., 1993. A numerical-simulation of flow in a 2-dimensional end-to-side anastomosis model. Journal of Biomechanical Engineering—Transactions of the ASME 115 (1), 112–118. Steinman, D.A., Thomas, J.B., Ladak, H.M., Milner, J.S., Rutt, B.K., Spence, J.D., 2002. Reconstruction of carotid bifurcation hemodynamics and wall thickness using computational fluid dynamics and MRI. Magnetic Resonance in Medicine 47 (1), 149–159. Sugiu, K., Kinugasa, K., Mandai, S., Tokunaga, K., Ohmoto, T., 1995. Direct thrombosis of experimental aneurysms with celluloseacetate polymer (cap)—technical aspects, angiographic follow- up, and histological study. Journal of Neurosurgery 83 (3), 531–538. Tabrizchi, R., Pugsley, M.K., 2000. Methods of blood flow measurement in the arterial circulatory system. Journal of Pharmacological and Toxicological Methods 44 (2), 375–384. Taylor, C.A., Draney, M.T., 2004. Experimental and computational methods in cardiovascular fluid mechanics. Annual Review of Fluid Mechanics 36, 197–231. Thompson, R.W., Liao, S.X., Curci, J.A., 1997. Vascular smooth muscle cell apoptosis in abdominal aortic aneurysms. Coronary Artery Disease 8 (10), 623–631. Tronc, F., Wassef, M., Esposito, B., Henrion, D., Glagov, S., Tedgui, A., 1996. Role of NO in flow-induced remodeling of the rabbit common carotid artery. Arteriosclerosis Thrombosis and Vascular Biology 16 (10), 1256–1262. Vinuela, F., Duckwiler, G., Mawad, M., Halbach, V., Berenstein, A., Dion, J., Graves, V., Hopkins, L.N., Ferguson, R., 1997. Guglielmi detachable coil embolization of acute intracranial aneurysm: perioperative anatomical and clinical outcome in 403 patients. Journal of Neurosurgery 86 (3), 475–482. Volin, M.V., Joseph, E., Shockley, M.S., Davies, P.F., 1998. Chemokine receptor CXCR4 expression in endothelium. Biochemical and Biophysical Research Communications 242 (1), 46–53. Wang, J.J., Parker, K.H., 2004. Wave propagation in a model of the arterial circulation. Journal of Biomechanics 37 (4), 457–470. Westerhof, N., Noordergraaf, A., 1970. Arterial viscoelasticity: a generalized model: effect on input impedance and wave travel in the systematic tree. Journal of Biomechanics 3 (3), 357–370. Wilkinson, I.B., Qasem, A., McEniery, C.M., Webb, D.J., Avolio, A.P., Cockcroft, J.R., 2002. Nitric oxide regulates local arterial distensibility in vivo. Circulation 105 (2), 213–217. Zarins, C.K., Giddens, D.P., Bharadvaj, B.K., Sottiurai, V.S., Mabon, R.F., Glagov, S., 1983. Carotid bifurcation atherosclerosis quantitative correlation of plaque localization with flow velocity profiles and wall shear-stress. Circulation Research 53 (4), 502–514. Zienkiewicz, O.C., 1971. The Finite Element Method in Engineering Science. McGraw-Hill Book Company, New York.
Journal of Biomechanics 40 (2007) 412–426 www.elsevier.com/locate/jbiomech www.JBiomech.com
Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model I. Chatziprodromoua, A. Tricolia, D. Poulikakosa, Y. Ventikosb, a
Laboratory of Thermodynamics in Emerging Technologies, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Zentrum, CH-8092 Zurich, Switzerland b Fluidics and Biocomplexity Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Accepted 14 December 2005
Abstract We have developed a computational simulation model for investigating an often postulated hypothesis connected with aneurysm growth. This hypothesis involves a combination of two parallel and interconnected mechanisms: according to the first mechanism, an endothelium-originating and wall shear stress-driven apoptotic behavior of smooth muscle cells, leading to loss of vascular tone is believed to be important to the aneurysm behavior. Vascular tone refers to the degree of constriction experienced by a blood vessel relative to its maximally dilated state. All resistance and capacitance vessels under basal conditions exhibit some degree of smooth muscle contraction that determines the diameter, and hence tone, of the vessel. The second mechanism is connected to the arterial wall remodeling. Remodeling of the arterial wall under constant tension is a biomechanical process of rupture, degradation and reconstruction of the medial elastin and collagen fibers. In order to investigate these two mechanisms within a computationally tractable framework, we devise mechanical analogues that involve three-dimensional haemodynamics, yielding estimates of the wall shear stress and pressure fields and a quasi-steady approach for the apoptosis and remodeling of the wall. These analogues are guided by experimental information for the connection of stimuli to responses at a cellular level, properly averaged over volumes or surfaces. The model predicts aneurysm growth and can attribute specific roles to the two mechanisms involved: the smooth muscle cell-related loss of tone is important to the initiation of aneurysm growth, but cannot account alone for the formation of fully grown sacks; the fiber-related remodeling is pivotal for the latter. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cerebral aneurysms; Growth model; Computational simulation; Biological fluid mechanics
1. Introduction The mechanics of cerebral saccular aneurysm pathogenesis are not yet well understood. Although different biological mechanisms are identified as possible reasons for the genesis, the growth and the aneurysm rupture, a fully descriptive theory that explains the relevant biomechanical mechanisms cannot be found in the literature. The aim of this study is to investigate computationally a hypothesis that is relating mechanical Corresponding author. Tel.: +44 1865 277944.
E-mail addresses: [email protected] (D. Poulikakos), [email protected] (Y. Ventikos). 0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2005.12.009
and biological responses of the arterial wall, with the hope of shedding some light to a possible biomechanically explained process. The vascular endothelium (VE) comprises the inner surface of blood vessels and its main role is to be a selective barrier between the blood and the rest of the biological tissues, as well as a wall shear stress sensor. The reaction of the vascular endothelium to blood flow was observed more than 150 years ago; Virchow identified that the morphology of the endothelial cells is varying along the arterial tree due to the variability of the corresponding blood flow patterns (Resnick et al., 2003). Since then, the underlying mechanisms that are responsible for the endothelial cell (EC) function
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
became an area of interest for many investigators. In this direction, a lot of different aspects, such as Ca++ and K+ ion channels activation (Adams et al., 1989; Ando et al., 1993), gene expression differentiation due to shear forces (Volin et al., 1998; Resnick et al., 1997; Shyy et al., 1994), mechanotransduction and mechanosignaling events (Davies et al., 1997; Helmke and Davies, 2002), reorganization and EC alignment (Malek and Izumo, 1996; Satcher et al., 1992), have been studied in depth for the past three decades. One of the most important tasks of the VE is to regulate the arterial wall properties through a blood shear stress activation mechanism. Luis J. Ignarro in his Nobel Prize lecture summarized the steps taken that led to the identification of NO as a principal endothelium derived relaxation factor (EDRF) (Ignarro, 1999). This endogenous molecule alters the properties of the arterial wall by relaxing the smooth muscle cells (SMC) embedded in the arterial media layer. A series of studies have shown that NO production is positively correlated with the changes of the shear stress (SS) values on the arterial wall through an endothelium-dependent biological mechanism (Buga et al., 1991; Tronc et al., 1996; Snow et al., 2001; Wilkinson et al., 2002; Butler et al., 2000; Kinlay et al., 2001). In our aneurysmal genesis hypothesis, this correlation plays a key role for the initialization of the lesion. The quantification of clinically important haemodynamic variables, such as blood velocity, pressure and shear stress, can provide information for the healthy and diseased cardiovascular function. There are numerous experimental and computational methods that are targeting the estimation of such quantities. In practice, parameters like shear stress and blood flow rate are derived indirectly from the primary parameters (pressure and velocity) that are measured/computed directly. Experimental methods can be divided in in-vitro methods (where phantoms are used) and in-vivo methods (where the experiments are conducted directly on human or animal subjects). In-vitro methods are considerably easier to employ, because they do not have the ethical and technological restrictions that in-vivo methods inevitably entail. Catheters equipped with tip pressure transducers are the main instrument for pressure measurements for both in-vitro (Millar and Baker, 1973) and in-vitro (Debruyne et al., 1994) experiments. As far as flow evaluation is concerned, a large number of qualitative and quantitative experimental techniques exist, like hydrogen bubble visualization (Zarins et al., 1983), transit time ultrasound flow probes (Tabrizchi and Pugsley, 2000; Molloi et al., 1996; Molloi et al., 2004), and hot wire velocimetry (Nerem and Seed, 1972). Advances in technology allowed the development of methods that are not invasive (or are minimally invasive). Such methods include computer tomography (CT), ultrasound and magnetic resonance
413
imaging (MRI). Ultrasound and MRI represent the current and prevailing methods in in vivo blood velocimetry but are still not always able to produce dependable and accurate measurements, due to motion artifacts, temporal and spatial resolution limitations, etc. Despite current limitations, these techniques show great promise for providing subject-specific details on important haemodynamic quantities. MRI and Ultrasound techniques find a complimentary counterpart in a modeling methodology developed in recent years, namely Computational Haemodynamics. The main advantages of computational techniques, is that they are non-invasive, they are fully repeatable and they are often capable of providing us with insightful information that is not accessible by any other means. The computational efforts went through a number of steps and advanced in parallel with the computer technology evolution. The result of this parallel evolution produced methods that are ranging from 0D to 3D models (Sherwin et al., 2000), idealized or realistic geometries, with deformable and/or collapsible (fluid-structure interaction) (Otis et al., 1993; Bertram and Pedley, 1982; Perktold and Rappitsch, 1995) or rigid walls and Newtonian or non-Newtonian (Ballyk et al., 1994) approximation for the blood. More specifically, as far as 0D and 1D models are concerned, lamped capacity models, making use of an analogy between fluid flow and electric circuits or other computer-time effective techniques (Westerhof and Noordergraaf, 1970; Wang and Parker, 2004), have yielded useful information. More detailed local information started becoming available with 2D (Steinman et al., 1993) and 3D models (Perktold et al., 1998; Rappitsch et al., 1997; Steiger and Perktold, 1997; Boutsianis et al., 2004) that resulted in a more descriptive perspective of the flow field. Furthermore, the MRI technology makes possible more sophisticated calculations within realistic and patient-specific arterial geometries (Butty et al., 2002; Steinman, 2002; Steinman et al., 2002; Chatziprodromou et al., 2003; Taylor and Draney, 2004). As mentioned before, computational techniques can be closely connected with Ultrasound or MRI blood velocimetry, since these methods provide boundary conditions for patient-specific haemodynamics. Finally, a topic that has drawn the attention of the bioengineering community, of interest to the present study, is the exploration of the structural properties and behavior of the arterial wall. In this direction, both experimental and computational methods have been published. Arterial residual stresses (Fung, 1991; Fung and Liu, 1992), elastic properties and behavior of the various arterial layers (Jogestrand et al., 2003) and mathematical modeling of the arterial function (SchulzeBauer and Holzapfel, 2003; Holzapfel et al., 2002; Gasser and Holzapfel, 2003) are some of the aspects that
ARTICLE IN PRESS 414
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
the scientists are focusing their efforts. A comprehensive review of continuum biomechanics for soft biological tissues can be found in Humphrey (2003). The two important biological mechanisms upon which our hypothesis is based are the apoptosis of the medial smooth muscle cells (Kondo et al., 1998; Schmid et al., 2003; Thompson et al., 1997; Sakaki et al., 1997) and the breakage and elimination of the collagen and elastin fibers within the aneurysmal wall (Mimata et al., 1997; Gaetani et al., 1998; Carmo et al., 2002; Finlay et al., 1995). The present work explores a hypothesis related to the inception and growth of saccular cerebral aneurysms. Our hypothesis is summarized as follows: Hypothesis. A VE malfunction or/and an abnormal shear stress field due to the presence of atherosclerotic plaque or an arterial bifurcation, can cause an overexpression of the Endothelium dependent NO production which leads to smooth muscle cell relaxation and consequently to lower, non-physiological local arterial aggregate Young’s Modulus of Elasticity. This process results in a disturbance of the equilibrium between the blood pressure forces and the internal wall stress forces in favor of the first and subsequently dilates locally the arterial wall causing an initial ballooning. The resulting blood shear stress field—after the above-described geometrical development—in conjunction with a possible thinning of the medial layer because of the smooth muscle cell apoptotic mechanism, are the driving forces for further growth of the aneurysmal geometry. This geometrical growth stretches the collagen and elastin fibers of the medial and adventitial layers and gives rise to internal stresses that contribute to the arterial stiffness. Eventually, the biomechanical system equilibrates at a state where the internal wall stresses and the transmural pressure are equal whilst the local haemodynamics cannot alter the arterial properties any more. At this point, the elastin and the collagen fibers that are responsible for the internal stresses are constantly under a non-physiological, large mechanical load. This tensile state of the above mentioned fibers gradually causes their degradation and breakage and results in a remodeling of the wall and consequently in a new vascular tone. In this new state, the artery is again vulnerable to the pressure field, since we have a new equilibrium and further aneurysmal growth, or rupture, is possible. In order to investigate this hypothesis, we have developed a model that includes the luminal haemodynamics and the tissue response within a computationally tractable framework. Inevitably, a number of assumptions are made, especially with respect to the extremely complicated and poorly understood biochemistry and biomechanics of the processes discussed. The model will be presented in the following section. Subsequently the
results from its application to an idealized aneurysm growth study will be discussed.
2. Modeling and computational methods The core of the model involves a quasi-steady approach, supported by the disparate scales of the local time-accurate haemodynamics, O (s), and the evolution of an aneurysm, O (months) or more. The approach entails the estimation of a prevalent haemodynamic state and subsequently the estimation of the evolution of the disease under the assumption that this state remains constant, until geometric considerations dictate a reevaluation of the haemodynamics. In this manner, we have the flexibility to account for the haemodynamics in a variety of ways. In the present study, and for expedience, we have conducted all the step-wise quasisteady computations of the haemodynamic behavior in a steady (average flow) manner. This approach serves in demonstrating the key features of the model, which lie in the decoupling of the scales and in the coupling of the mechanics with the cellular responses, but it is not restrictive. An idealized geometry of a common carotid artery was used for the application of the computational model that mimics the process described in our hypothesis. The geometry was constructed with the use of a CAD modeler/grid generator (Gambit, FLUENT INC.), Fig. 1(a). This initial geometry was used for a stress pre-evaluation step: Arteries at normal equilibrium conditions have residual stresses in the arterial walls, because of the blood flow and the pressure of the blood they contain. In order to model these stresses, we performed an initial calculation, in a fully coupled fluidstructure interaction manner that was equivalent with filling an unstressed (i.e. empty) artery with blood at physiological flow conditions. The resulting geometry of this initial computation with the calculated internal stresses was used as the base for all further calculations (Fig. 1(b)). This first step caused an increase of the initial diameter of the common carotid artery from 6.5 mm and a corresponding Reynolds number of 200 to a diameter of 8 mm and a Reynolds equal to 165. The thickness of the arterial wall was 0.75 mm and the length of the tubular model was 65 mm. The final dimensions and flow conditions achieved are intended to be typical for carotids. A Young’s modulus of Elasticity of 120 000 Pa and a Poisson ratio of 0.4 were used for this initial step. For the blood, a Newtonian behavior assumption was made (Perktold et al., 1991; Cho and Kensey, 1991). Constant density of 1069 kg/m3 and dynamic viscosity coefficient of 0.0035 (Kg/m s) were the blood physical parameters. The boundary conditions employed were uniform velocity equal to 0.1 m/s for the inlet and fixed physiological pressure 6665 Pa (50 mm Hg) for the
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
415
Fig. 1. Flow and Stress Boundary Conditions and resulting geometry after the internal stress pre-evaluation step.
Fig. 3. Volume region nomenclature.
Fig. 2. Wall Strain rate on a line along the X-axis (vessel axial line) that passes from the top of the aneurysm for the time t6a (the most distorted geometry computed). Two grids compared: 395094 elements and 833975 elements.
outlet. These boundary conditions correspond to a characteristic flow setting for typical carotid arteries. The boundary conditions that were used for the solid problem were fixed zero displacement for the face adjacent to the inflow and outflow sections and a free boundary condition for the outer surface of the arterial wall. The coupling of the fluid mechanics and the solid deformations was achieved through an implicit pressure boundary condition at the luminal arterial wall. After this initial step was completed, the general quasi-steady algorithm was employed to study the growth of an induced aneurysm in this pre-stressed arterial segment. This algorithm can be summarized as follows: 1. An adequate (after a grid independence study, Fig. 2) unstructured tetrahedral grid was constructed. The grid consisted of 395094 elements for the flow
solution computations and of 915179 elements for the fluid-structure interaction runs (444223 elements for the fluid volume—470956 elements for the solid volume). At each time step, the mesh was refined and the element size kept constant, in spite of the deformations in the geometry. The grid was generated using a mesh generator (Gambit, Fluent) for the initial regional constituents. 2. A finite volume solver was employed for the numerical solution of the flow within the parent vessel and the aneurysm (CFD–ACE+, ESI CFD). A second order accurate discretization scheme was used, along with a SIMPLEC-type pressure-velocity coupling scheme. Algebraic multigrid acceleration was employed in all simulations in order to enhance the speed of convergence of the solution. The blood flow equations that were solved by the finite volume solver are the incompressible Navier–Stokes equations along with continuity, i.e. the conservation laws for momentum and mass for incompressible fluids. The conservation of mass can be expressed as @ui ¼ 0. @xi
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
416
Conservation of momentum reads: r
@ui @ui @p @tij þ ruj ¼ þ þ SMx . @xi @xj @t @xj
The viscous stress components tij read: 1 @ui @uj tij ¼ 2msij where sij ¼ þ 2 @xj @xi for Newtonian flows with constant viscosity m. 3. Through a fluid-structure interaction process, the flow-induced forces that were computed act as an Table 1 Geometrical data for the two volumes where the geometrical evolution took place
Ellipsoidal Ring shaped
R1 (m, big axis of the ellipse along the length of the tube)
R2 (m, small axis of the ellipse perpendicular to the length of the tube)
0.0035 0.0045
0.002 0.003
implicit force on the arterial wall. The resulting wall deformation was calculated using a finite element stress solver (CFD–ACE+, ESI CFD). This fluidstructure interaction computation is based on the strong implicit coupling between the solid-stress strain computation and the fluid computation. Pressure and shear stress information is sent to the stress computation module, where deformations and stresses are calculated. Subsequently, these deformations are fed back to the fluid module, where a new grid is generated and the solution is recalculated on this new deformed geometry. Iterations are performed until convergence is obtained (i.e. the residuals for the three velocity components and for the pressure dropped nine orders of magnitude and stabilized). The structural part of this algorithm is based on the Finite Element method. The principle of virtual work (Zienkiewicz, 1971) is used for the implementation of the stress-deformation computation. For each element, displacements are defined at the nodes and obtained within the element by interpolation from the nodal values using appropriate
Fig. 4. Geometrical evolution contours due to the Young’s Modulus of Elasticity dynamic alteration on the longitudinal symmetry plane.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
shape functions. Using the strain–displacement relationship, the strains e are derived from the displacements u, hence the nodal displacements a. This may be expressed as de ¼ Bda. If the displacements are large, the strains depend nonlinearly on the displacements, thus B is a function of a. This is expressed as B ¼ B0 þ BL ðaÞ, where B0 is the standard small-strain strain–displacement matrix, and BL is a linear function of the nodal displacement. The governing equations are derived by forming a balance between the external and internal generalized forces using the principal of virtual work mentioned above. Equating the external work done with the total internal work, and recognizing this equality must be valid for any value of virtual displacement, we reach an equilibrium equation which for the general nonlinear case is solved iteratively via a Newton–Raphson technique. We must note that, although the general elasticity relationship, s ¼ Dðe2e0 Þ2s0 , is used, this approach is sufficiently general to allow for any nonlinear stress–strain relationship, even the
417
variable exponential ones that prevail in artery wall dynamics, since the solution will again reduce to the solution of a set of nonlinear equations, (Bathe and Wilson, 1976; Newmark, 1959; Roark and Young, 1975). We approximated the arterial wall as a fully elastic medium, with the above-mentioned characteristics. 4. The above step causes a geometrical evolution and a subsequent grid deformation both for the fluid and the solid areas. Smooth and controlable deformation of the luminal zone was derived by solving a pseudostress-strain problem for the fluid volume, geared exclusively towards evaluating the new position of the internal nodes under the new boundary node configuration. When the deformed mesh proved to be of sufficient quality for the re-calculation of the flow within the new geometry, that computation took place in a straightforward manner. In situations were the deformation was too large, remeshing of the luminal volume was conducted, to satisfy the abovementioned constraint.
Fig. 5. Geometrical evolution contours due to the Young’s Modulus of Elasticity dynamical alteration up to time t5 and the arterial adaptation that is taking place at time t6a.
ARTICLE IN PRESS 418
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
The geometrical evolution was carried out in eight quasi-steady steps. For inducing the inception of an aneurysm-prone area, a simplified two-zonal differentiation was imposed to the wall properties: An ellipsoidal (EL) and a ring-shaped (RS) zone were introduced,
(Fig. 3 and Table 1). At these regions, the strength of the artery (i.e. YME) was decreased linearly (due to the lack of quantitative data) during the computation of the aneurysm evolution. This linear dynamic YME alteration is used to model the above described endothelial cell
Fig. 6. Wall Shear Stress patterns in (Pa) for the eight time instances at the aneurysm region (top view).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
layer function and the corresponding smooth muscle cell degeneration which is derived from a combination of information found in numerous, predominantly in vitro, investigations (Ignarro, 1999; Buga et al., 1991; Tronc
419
et al., 1996; Snow et al., 2001; Wilkinson et al., 2002; Butler et al., 2000; Kinlay et al., 2001). This change in the YME for the EL was in the range from 120 000 to 20 000 Pa, while the corresponding line for the RS
Fig. 7. Pressure patterns in (Pa) for the eight time instances at the aneurysm region (top view).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
420
ranged between the values 120 000 and 40 000 Pa. The reasoning for this differentiation of the two regions stems from the fact that the aneurysmal neck has been shown to develop significantly higher values of WSS (Tronc et al., 1996; Gnasso et al., 2001). The above-mentioned eight consecutive steps of wall weakening allow us to draw interesting conclusions on the plausibility of the mechanical wall-weakening hypothesis, as we shall see in the next section. To fully explore our basic hypothesis, one additional mechanism has to be incorporated though: the effect of arterial wall remodeling under constant load, a mechanism is of major importance as we shall see. There is inconclusive evidence in the literature as to whether this process occurs rapidly (always referring to disease evolutiontime terms) or gradually. For this study, we decided to simulate this behavior under the (strong) assumption that the aneurysmal wall fully remodels itself to an unstressed state within one simulated disease evolution step. The reason for this assumption is that we would like at this stage to point out the importance of the remodeling to the aneurysmal evolution. The abruptness involved in the rupture of individual collagen fibers (Mimata et al., 1997; Gaetani et al., 1998; Carmo et al., 2002; Finlay et al., 1995) (a process that in principle justifies our assumption) is smoothened out in time by the fact that there are numerous fibers undergoing this rupture and re-adjustment of tensile state. The stress readjustment calculation was performed on the geometry that was resulted from the 5th disease evolution step (t5) calculation.
3. Results and discussion Fig. 4 shows the evolution of the aneurysmal shape only due to the dynamic alteration of the Young’s modulus of elasticity. If we consider that the final YME values for time t ¼ 9 of the ring- and the ellipsoidshaped regions are highly abnormal (20 000 and 40 000 Pa, respectively) when at the same time the displacement is not so significant, (compared to cases of giant aneurysms, (Vinuela et al., 1997)) we can conclude that this mechanism alone is not enough to
describe the aneurysmal evolution. In all likelihood, this mechanical process is reflecting a primary malfunction of the endothelial cell layer and the corresponding loss of vascular tone. Moreover, although there exists evidence of apoptosis of the smooth muscle cells during the growth of the aneurysm, we do not know why and how exactly this takes place, (Kondo et al., 1998; Schmid et al., 2003; Thompson et al., 1997; Sakaki et al., 1997). Definitive evidence that the endothelial cell activity is directly connected with the apoptotic processes is missing. In effect, this loss of tone is assumed to account for the initialization of the lesion. This comment is in agreement with the observed high concentration of endothelial cells at high shear stress regions (Hoshina et al., 2003) while histological data indicate that the endothelial cell layer within the aneurysmal lumen is quite normal (Sugiu et al., 1995). This conclusion would indicate that, while we have an unusually high concentration of endothelial cells at the early stages of the decease, at the later stages we have a normal alignment of the cells with respect to the inner layer of the aneurysmal lumen. The YME dynamical evolution up to time t ¼ 5, along with the arterial adaptation mechanism (i.e. canceling of the internal stresses values and establishment of a stress- free state for the resulted geometry from the t ¼ 5 simulation) that took place at time 6a resulted in the sequence of the geometries that are depicted in Fig. 5. It is obvious that the resulting geometry for t ¼ 6a is significantly more deformed than the one achieved by the simple loss of tone/apoptosis mechanism. The exact magnitude of this particular deformation is of course also dependent on the fact that a total loss of stresses was assumed in the fibers of the wall during this step, an assumption that, as we Table 3 Total integral pressure force on the various segments of the inner (in) surface of the arterial wall (N)
Fine
Stretched
Wall in Ring in Elips. in
0.370001 0.175 0.194
0.370001 0.175 0.194
Table 2 Integral pressure force components on the various segments of the inner (in) surface of the arterial wall (N)
X
Y
Fine Wall in Ring in Elips. in
Stretched 4
6.92 10 6.37 105 1.41 104
Z
Fine 4
7.05 10 6.36 105 1.41 104
Stretched 1
3.70 10 1.75 101 1.94 101
Fine 1
3.70 10 1.75 101 1.94 101
Stretched 5
5.69 10 5.49 105 1.77 106
5.68 105 5.44 105 2.05 106
Stretched grid: The grid that results from the grid deformation due to the geometrical evolution. Fine grid: The grid-independent regenerated mesh that was used for the blood flow calculation.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
discussed, is useful for demonstration purposes but possibly not entirely realistic. The influence and the importance of the adaptation procedure will be examined in the future, in correlation with availability of micro-histology data. The Wall Shear Stress patterns at the various stages during the aneurysmal growth are shown in Fig. 6. These results coincide and justify the observation that the shear stress presents with higher values at the neck neighborhood, hence, the YME is changing more than the corresponding value at the center of the lumen at each time step (Banerjee et al., 1996). Essentially, mean WSS values identified in this step are the driving stimuli that dictate the variation of the YME of the corresponding layers of the arterial wall. The bi-zonal approach we
421
developed is also justified by the average observed behavior of the WSS in these figures. Fig. 7 shows that the pressure field on the wall remains practically constant for the entire domain. This result, together with the fact that the pressure field depicted is obtained after the blood lumen grid is adequately refined, confirms the statement that the coupled flow-stress calculation is sufficiently resolved for driving the process (Tables 2 and 3). The streamline (color-coded with the velocity magnitude) comparison between the t ¼ 6 and 6a case is presented in Fig. 8. The importance of the arterial wall adaptation mechanism and its influence on the arterial haemodynamics is further accentuated by this comparison. The latter case leads to a severely deformed geometry faster; the flow field
Fig. 8. Wall Shear Stress and Streamline comparison for the corresponding geometries at the time instances t6 and t6a (resulting geometry after the wall remodelling and the ballooning from the fluid—structure interaction).
ARTICLE IN PRESS 422
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
within the aneurysmal sac is much more disturbed at this stage. This in turn results in a wall shear stress spatial distribution of more irregular patterns and the possibility of further activation of the endothelial cell function.
Since the arterial wall stress tensor has six components and it is difficult to present any illustrative results based on it, the energetic equivalent formulation by von Mises is used to depict the stress history of the arterial
Fig. 9. Von Mises stresses in (N/m2) for the eight time instances at the aneurysm region on the inner surface of the arterial wall.
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
wall during its deformation process. The equivalent von Mises stress characterizes the distortional energy (s2/E) within an element and gives a stress value in terms of equivalent positive defined scalar stress levels. In Figs. 9 and 10 contours of the above-mentioned quantity are shown, for the luminal and adventitial surfaces of the arterial wall respectively. The symmetry
423
of the geometry is reflected on the von Mises stress values for both the inner and the outer surfaces. The total absence of initial internal stresses for the wall remodeling case causes an abrupt and rapid deformation (Fig. 11). The difference between the final stresses of the remodeled and the non-remodeled case is clear. This difference is a result of our decision to eliminate the
Fig. 10. Von Mises stresses in (N/m2) for the eight time instances at the aneurysm region on the outer surface of the arterial wall.
ARTICLE IN PRESS 424
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
internal stresses in order to underline the importance of the remodeling procedure. This result gives the impression that the bulk part of the deformation is caused by this remodeling procedure and more or less the shaping of the aneurysm is the result of an one step procedure. From Fig. 11, we can also see that the resulting spatial stresses from this deformation are similar to that during the inflation of a balloon. Moreover, because of the large deformation near the apex of this ‘‘balloon’’, we are finding the largest stress values for the outer wall at the territory with the higher YME value.
4. Conclusions We have developed a computational model for the growth of saccular cerebral aneurysms. The model relies
on certain simplifying assumptions: the bizonal approach, and the relation of smooth muscle cell tone loss and apoptosis with the WSS patterns are two of the most important ones. Moreover, for the purposes of the present study, we have limited the driving haemodynamics to steady state simulations. On the other hand, correlating accurately detailed quantitative data of vasoactive agent overproduction with local wall shear stress patterns involves detailed in vitro or better yet in vivo experiments which, to the best of our knowledge, do not exist. With the above approximations, the model developed shows that the combination of two individual mechanisms, often postulated in the literature as responsible for aneurysm growth, are indeed of importance to the process: Loss of vascular tone due to smooth muscle cell apoptosis and reconstitution of fibers after prolonged periods of excessive strain have
Fig. 11. Von Mises stresses (N/m2) comparison on the inner and the outer surfaces of the arterial wall, between the corresponding geometries at the time points t6 and t6a (resulting geometry after the wall remodelling and the ballooning from the fluid—structure interaction).
ARTICLE IN PRESS I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
been shown to contribute to the condition. Moreover, we have provided evidence that although the first mechanism is important to the inception of the disease, for substantial aneurysm growth to occur, the second mechanism is pivotal.
Acknowledgements The authors are grateful to the ETH Forschungscommission for supporting this work. The ESI Group and CFD–RC (Dr. S. Sundaram) are kindly acknowledged for their advice and support and for allowing the usage of the CFD–ACE platform. One of the authors (YV) would like to acknowledge discussions with Dr. Robin Poston of King’s College, London, that affirmed further our causal hypothesis for the disease, based on pathological and histological data, during the revision of this paper. References Adams, D.J., Barakeh, J., Laskey, R., Vanbreemen, C., 1989. Ion channels and regulation of intracellular calcium in vascular endothelial-cells. Faseb Journal 3 (12), 2389–2400. Ando, J., Ohtsuka, A., Korenaga, R., Kawamura, T., Kamiya, A., 1993. Wall shear-stress rather than shear rate regulates cytoplasmic Ca++ responses to flow in vascular endothelial-cells. Biochemical and Biophysical Research Communications 190 (3), 716–723. Ballyk, P.D., Steinman, D.A., Ethier, C.R., 1994. Simulation of nonnewtonian blood-flow in an end-to-side anastomosis. Biorheology 31 (5), 565–586. Banerjee, R.K., Gonzalez, C.F., Cho, Y.I., Picard, L., 1996. Hemodynamic changes in recurrent intracranial terminal aneurysm after endovascular treatment. Academic Radiology 3 (3), 202–211. Bathe, K., Wilson, E.L., 1976. Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, New Jersey. Bertram, C.D., Pedley, T.J., 1982. A mathematical-model of unsteady collapsible tube behavior. Journal of Biomechanics 15 (1), 39–50. Boutsianis, E., Dave, H., Frauenfelder, T., Poulikakos, D., Wildermuth, S., Turina, M., Ventikos, Y., Zund, G., 2004. Computational simulation of intracoronary flow based on real coronary geometry. European Journal of Cardio-Thoracic Surgery 26 (2), 248–256. Buga, G.M., Gold, M.E., Fukuto, J.M., Ignarro, L.J., 1991. Shearstress induced release of nitric-oxide from endothelial-cells grown on beads. Hypertension 17 (2), 187–193. Butler, P.J., Weinbaum, S., Chien, S., Lemons, D.E., 2000. Endothelium-dependent, shear-induced vasodilation is rate- sensitive. Microcirculation 7 (1), 53–65. Butty, V.D., Gudjonsson, K., Buchel, P., Makhijani, V.B., Ventikos, Y., Poulikakos, D., 2002. Residence times and basins of attraction for a realistic right internal carotid artery with two aneurysms. Biorheology 39 (3–4), 387–393. Carmo, M., Colombo, L., Bruno, A., Corsi, F.R.M., Roncoroni, L., Cuttin, M.S., Radice, F., Mussini, E., Settembrini, P.G., 2002. Alteration of elastin, collagen and their cross-links in abdominal aortic aneurysms. European Journal of Vascular and Endovascular Surgery 23 (6), 543–549. Chatziprodromou, I., Butty, V.D., Makhijani, V.B., Poulikakos, D., Ventikos, Y., 2003. Pulsatile blood flow in anatomically accurate vessels with multiple aneurysms: a medical intervention planning
425
application of computational haemodynamics. Flow Turbulence and Combustion 71 (1–4), 333–346. Cho, Y.I., Kensey, K.R., 1991. Effects of the non-newtonian viscosity of blood on flows in a diseased arterial vessel.1. Steady flows. Biorheology 28 (3–4), 241–262. Davies, P.F., Barbee, K.A., Volin, M.V., Robotewskyj, A., Chen, J., Joseph, L., Griem, M.L., Wernick, M.N., Jacobs, E., Polacek, D.C., DePaola, N., Barakat, A.I., 1997. Spatial relationships in early signaling events of flow-mediated endothelial mechanotransduction. Annual Review of Physiology 59, 527–549. Debruyne, B., Stockbroeckx, J., Demoor, D., Heyndrickx, G.R., Kern, M.J., 1994. Role of side holes in guide catheters—observations on coronary pressure and flow. Catheterization and Cardiovascular Diagnosis 33 (2), 145–152. Finlay, H.M., McCullough, L., Canham, P.B., 1995. 3-Dimensional collagen organization of human brain arteries at different transmural pressures. Journal of Vascular Research 32 (5), 301–312. Fung, Y.C., 1991. What are the residual-stresses doing in our bloodvessels. Annals of Biomedical Engineering 19 (3), 237–249. Fung, Y.C., Liu, S.Q., 1992. Strain distribution in small blood-vessels with zero-stress state taken into consideration. American Journal of Physiology 262 (2), H544–H552. Gaetani, P., Tartara, F., Grazioli, V., Tancioni, F., Infuso, L., Baena, R.R.Y., 1998. Collagen cross-linkage, elastolytic and collagenolytic activities in cerebral aneurysms: a preliminary investigation. Life Sciences 63 (4), 285–292. Gasser, T.C., Holzapfel, G.A., 2003. Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Computer Methods in Applied Mechanics and Engineering 192 (47–48), 5059–5098. Gnasso, A., Carallo, C., Irace, C., De Franceschi, M.S., Mattioli, P.L., Motti, C., Cortese, C., 2001. Association between wall shear stress and flow-mediated vasodilation in healthy men. Atherosclerosis 156 (1), 171–176. Helmke, B.P., Davies, P.F., 2002. The cytoskeleton under external fluid mechanical forces: Hemodynamic forces acting on the endothelium. Annals of Biomedical Engineering 30 (3), 284–296. Holzapfel, G.A., Gasser, T.C., Stadler, M., 2002. A structural model for the viscoelastic behavior of arterial walls: continuum formulation and finite element analysis. European Journal of Mechanics aSolids 21 (3), 441–463. Hoshina, K., Sho, E., Sho, M., Nakahashi, T.K., Dalman, R.L., 2003. Wall shear stress and strain modulate experimental aneurysm cellularity. Journal of Vascular Surgery 37 (5), 1067–1074. Humphrey, J.D., 2003. Continuum biomechanics of soft biological tissues. Proceedings of the Royal Society of London Series aMathematical Physical and Engineering Sciences 459 (2029), 3–46. Ignarro, L.J., 1999. Nitric oxide: a unique endogenous signaling molecule in vascular biology (Nobel lecture). Angewandte ChemieInternational Edition 38 (13–14), 1882–1892. Jogestrand, T., Eiken, O., Nowak, J., 2003. Relation between the elastic properties and intima-media thickness of the common carotid artery. Clinical Physiology and Functional Imaging 23 (3), 134–137. Kinlay, S., Creager, M.A., Fukumoto, M., Hikita, H., Fang, J.C., Selwyn, A.P., Ganz, P., 2001. Endothelium-derived nitric oxide regulates arterial elasticity in human arteries in vivo. Hypertension 38 (5), 1049–1053. Kondo, S., Hashimoto, N., Kikuchi, H., Hazama, F., Nagata, I., Kataoka, H., 1998. Apoptosis of medial smooth muscle cells in the development of saccular cerebral aneurysms in rats. Stroke 29 (1), 181–188. Malek, A.M., Izumo, S., 1996. Mechanism of endothelial cell shape change and cytoskeletal remodeling in response to fluid shear stress. Journal of Cell Science 109, 713–726.
ARTICLE IN PRESS 426
I. Chatziprodromou et al. / Journal of Biomechanics 40 (2007) 412–426
Millar, H.D., Baker, L.E., 1973. Stable ultraminiature catheter-tip pressure transducer. Medical & Biological Engineering 11 (1), 86–89. Mimata, C., Kitaoka, M., Nagahiro, S., Iyama, K., Hori, H., Yoshioka, H., Ushio, Y., 1997. Differential distribution and expressions of collagens in the cerebral aneurysmal wall. Acta Neuropathologica 94 (3), 197–206. Molloi, S., Ersahin, A., Tang, J., Hicks, J., Leung, C.Y., 1996. Quantification of volumetric coronary blood flow with dual- energy digital subtraction angiography. Circulation 93 (10), 1919–1927. Molloi, S., Zhou, Y.F., Kassab, G.S., 2004. Regional volumetric coronary blood flow measurement by digital angiography: in vivo validation. Academic Radiology 11 (7), 757–766. Nerem, R.M., Seed, W.A., 1972. In-vivo study of aortic flow disturbances. Cardiovascular Research 6 (1), 1–14. Newmark, N.M., 1959. A method of computation for structural dynamics. ASCE Journal of the Engineering Mechanics Division 85, 67–94. Otis, D.R., Johnson, M., Pedley, T.J., Kamm, R.D., 1993. Role of pulmonary surfactant in airway-closure—a computational study. Journal of Applied Physiology 75 (3), 1323–1333. Perktold, K., Rappitsch, G., 1995. Computer-simulation of local blood-flow and vessel mechanics in a compliant carotid-artery bifurcation model. Journal of Biomechanics 28 (7), 845–856. Perktold, K., Peter, R.O., Resch, M., Langs, G., 1991. Pulsatile nonnewtonian blood-flow in 3-dimensional carotid bifurcation models—a numerical study of flow phenomena under different bifurcation angles. Journal of Biomedical Engineering 13 (6), 507–515. Perktold, K., Hofer, M., Rappitsch, G., Loew, M., Kuban, B.D., Friedman, M.H., 1998. Validated computation of physiologic flow in a realistic coronary artery branch. Journal of Biomechanics 31 (3), 217–228. Rappitsch, G., Perktold, K., Pernkopf, E., 1997. Numerical modelling of shear-dependent mass transfer in large arteries. International Journal for Numerical Methods in Fluids 25 (7), 847–857. Resnick, N., Yahav, H., Khachigian, L.M., Collins, T., Anderson, K.R., Dewey, F.C., Gimbrone, M.A., 1997. Endothelial gene regulation by laminar shear stress. In: Analytical and Quantitative Cardiology, pp. 155–164. Resnick, N., Yahav, H., Shay-Salit, A., Shushy, M., Schubert, S., Zilberman, L.C.M., Wofovitz, E., 2003. Fluid shear stress and the vascular endothelium: for better and for worse. Progress in Biophysics & Molecular Biology 81 (3), 177–199. Roark, R.J., Young, W.C., 1975. In: M.-H.B. Co. (Ed.), Formulas for Stress and Strain, New York. Sakaki, T., Kohmura, E., Kishiguchi, T., Yuguchi, T., Yamashita, T., Hayakawa, T., 1997. Loss and apoptosis of smooth muscle cells in intracranial aneurysms—studies with in situ DNA end labeling and antibody against single-stranded DNA. Acta Neurochirurgica 139 (5), 469–474. Satcher, R.L., Bussolari, S.R., Gimbrone, M.A., Dewey, C.F., 1992. The distribution of fluid forces on model arterial endothelium using computational fluid-dynamics. Journal of Biomechanical Engineering—Transactions of the ASME 114 (3), 309–316. Schmid, F.X., Bielenberg, K., Schneider, A., Haussler, A., Keyser, A., Birnbaum, D., 2003. Ascending aortic aneurysm associated with bicuspid and tricuspid aortic valve: involvement and clinical relevance of smooth muscle cell apoptosis and expression of cell death- initiating proteins. European Journal of Cardio-Thoracic Surgery 23 (4), 537–543. Schulze-Bauer, C.A.J., Holzapfel, G.A., 2003. Determination of constitutive equations for human arteries from clinical data. Journal of Biomechanics 36 (2), 165–169. Sherwin, S.J., Shah, O., Doorly, D.J., Peiro, J., Papaharilaou, Y., Watkins, N., Caro, C.G., Dumoulin, C.L., 2000. The influence of
out-of-plane geometry on the flow within a distal end-to-side anastomosis. Journal of Biomechanical Engineering-Transactions of the Asme 122 (1), 86–95. Shyy, Y.J., Hsieh, H.J., Usami, S., Chien, S., 1994. Fluid Shear-Stress Induces a Biphasic Response of Human Monocyte Chemotactic Protein-1 Gene-Expression in Vascular Endothelium. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 91(11). pp. 4678–4682. Snow, H.M., Markos, F., O’Regan, D., Pollock, K., 2001. Characteristics of arterial wall shear stress which cause endotheliumdependent vasodilatation in the anaesthetized dog. Journal of Physiology—London 531 (3), 843–848. Steiger, H.J., Perktold, K., 1997. Computer modeling of intracranial saccular and lateral aneurysms for the study of their hemodynamics. Neurosurgery 41 (1), 326. Steinman, D.A., 2002. Image-based computational fluid dynamics modeling in realistic arterial geometries. Annals of Biomedical Engineering 30 (4), 483–497. Steinman, D.A., Vinh, B., Ethier, C.R., Ojha, M., Cobbold, R.S.C., Johnston, K.W., 1993. A numerical-simulation of flow in a 2-dimensional end-to-side anastomosis model. Journal of Biomechanical Engineering—Transactions of the ASME 115 (1), 112–118. Steinman, D.A., Thomas, J.B., Ladak, H.M., Milner, J.S., Rutt, B.K., Spence, J.D., 2002. Reconstruction of carotid bifurcation hemodynamics and wall thickness using computational fluid dynamics and MRI. Magnetic Resonance in Medicine 47 (1), 149–159. Sugiu, K., Kinugasa, K., Mandai, S., Tokunaga, K., Ohmoto, T., 1995. Direct thrombosis of experimental aneurysms with celluloseacetate polymer (cap)—technical aspects, angiographic follow- up, and histological study. Journal of Neurosurgery 83 (3), 531–538. Tabrizchi, R., Pugsley, M.K., 2000. Methods of blood flow measurement in the arterial circulatory system. Journal of Pharmacological and Toxicological Methods 44 (2), 375–384. Taylor, C.A., Draney, M.T., 2004. Experimental and computational methods in cardiovascular fluid mechanics. Annual Review of Fluid Mechanics 36, 197–231. Thompson, R.W., Liao, S.X., Curci, J.A., 1997. Vascular smooth muscle cell apoptosis in abdominal aortic aneurysms. Coronary Artery Disease 8 (10), 623–631. Tronc, F., Wassef, M., Esposito, B., Henrion, D., Glagov, S., Tedgui, A., 1996. Role of NO in flow-induced remodeling of the rabbit common carotid artery. Arteriosclerosis Thrombosis and Vascular Biology 16 (10), 1256–1262. Vinuela, F., Duckwiler, G., Mawad, M., Halbach, V., Berenstein, A., Dion, J., Graves, V., Hopkins, L.N., Ferguson, R., 1997. Guglielmi detachable coil embolization of acute intracranial aneurysm: perioperative anatomical and clinical outcome in 403 patients. Journal of Neurosurgery 86 (3), 475–482. Volin, M.V., Joseph, E., Shockley, M.S., Davies, P.F., 1998. Chemokine receptor CXCR4 expression in endothelium. Biochemical and Biophysical Research Communications 242 (1), 46–53. Wang, J.J., Parker, K.H., 2004. Wave propagation in a model of the arterial circulation. Journal of Biomechanics 37 (4), 457–470. Westerhof, N., Noordergraaf, A., 1970. Arterial viscoelasticity: a generalized model: effect on input impedance and wave travel in the systematic tree. Journal of Biomechanics 3 (3), 357–370. Wilkinson, I.B., Qasem, A., McEniery, C.M., Webb, D.J., Avolio, A.P., Cockcroft, J.R., 2002. Nitric oxide regulates local arterial distensibility in vivo. Circulation 105 (2), 213–217. Zarins, C.K., Giddens, D.P., Bharadvaj, B.K., Sottiurai, V.S., Mabon, R.F., Glagov, S., 1983. Carotid bifurcation atherosclerosis quantitative correlation of plaque localization with flow velocity profiles and wall shear-stress. Circulation Research 53 (4), 502–514. Zienkiewicz, O.C., 1971. The Finite Element Method in Engineering Science. McGraw-Hill Book Company, New York.